308 ANALYTICAL MECHANICS path with a period equal to . The axes of the path coin- co cide with the coordinate axes. Case IV. When d = ± - and 6 = a; the path becomes a cir- z> cle, and the motion uniform circular motion with a period 1 4. 27T equal to - CO PROBLEMS. Find the resultant motion due to the superposition of the motions defined by the following equations: (1) x a cos o}t and y = a sin at. (2) x a cos (t + tQ) and y = a sin ~~ t. (3) re = a sin coi and T/ = a cos (£ + tQ). (4) a; = a sin (art + 5) and y = a cos (art 5). (5) £ = a cos art and ?/ = 6 sin art. (6) a; = a sin (art 5) and y = 6 cos (art + 5). (7) x = a sin (art ~) and y = 6 sin (ort + -) \ */ \ ^/ (8) £ = acos(coi + ) and y = 6sin (coi )- \ «V \ «V (9) ^ = a cos (ut j) and y = 6 cosf (10) .-r = a cos (art + 50 and ?/ = 6 sin (co^ + 5o). 241. Physical Pendulum. Any rigid body which is free to oscillate under the action of its own weight is called a physical or a compound pendulum. Let A, Fig. 137, be a rigid body which is free to oscillate about a horizontal axis through the point 0 and perpendicular to the plane of the paper. Further let c denote the position of the center of mass and D its distance from the axis. Then the torque equation gives /- = mgDsmd, (X) at where m is the mass of the body and 0 the angular displacement from the position of equilibrium.